order	256
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂×C₈² · Schur multiplier (H₂) = C₂²×C₈

Aliases

direct product	C₈×C₄⋊C₈
unambiguous semidirect product	C₄⋊C₈²

Subgroups

275 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 20 points:

(5,6,7,9,8,10,11,12);(2,4)(13,14,15,17,16,18,19,20);(1,2,3,4)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1247);